Cycle index of direct product of permutation groups and number of equivalence classes of subsets of Zv

AbstractA permutation group A of degree n acting on a set X has a certain number of orbits, each a subset of X. More generally, A also induces an equivalence relation on X(k) the set of all k subsets of X, and the resulting equivalence classes are called k orbits of A, or generalized orbits. A self-complementary k-orbit is one in which for every k-subset S in it, X—S is also in it. Our main results are two formulas for the number s(A) of self-complementary generalized orbits of an arbitrary permutation group A in terms of its cycle index. We show that self-complementary graphs, digraphs, and relations provide special classes of self-complementary generalized orbits.

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Cycle Index, Weight Enumerator, and Tutte Polynomial

The Electronic Journal of Combinatorics ◽

10.37236/1663 ◽

2002 ◽

Vol 9 (1) ◽

Cited By ~ 7

Author(s):

Peter J. Cameron

Keyword(s):

Permutation Group ◽

Linear Code ◽

Tutte Polynomial ◽

Weight Enumerator ◽

Permutation Groups ◽

Cycle Index ◽

Transitive Groups ◽

Index Weight

With every linear code is associated a permutation group whose cycle index is the weight enumerator of the code (up to a trivial normalisation). There is a class of permutation groups (the IBIS groups) which includes the groups obtained from codes as above. With every IBIS group is associated a matroid; in the case of a group from a code, the matroid differs only trivially from that which arises directly from the code. In this case, the Tutte polynomial of the code specialises to the weight enumerator (by Greene's Theorem), and hence also to the cycle index. However, in another subclass of IBIS groups, the base-transitive groups, the Tutte polynomial can be derived from the cycle index but not vice versa. I propose a polynomial for IBIS groups which generalises both Tutte polynomial and cycle index.

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Equivalence classes of functions on finite sets

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171282000696 ◽

1982 ◽

Vol 5 (4) ◽

pp. 745-762

Author(s):

Chong-Yun Chao ◽

Caroline I. Deisher

Keyword(s):

Equivalence Class ◽

Equivalence Classes ◽

Permutation Groups ◽

Weak Equivalence ◽

Finite Sets ◽

The Family ◽

Polya's Theorem ◽

Finite Set ◽

Pólya’S Theorem

By using Pólya's theorem of enumeration and de Bruijn's generalization of Pólya's theorem, we obtain the numbers of various weak equivalence classes of functions inRDrelative to permutation groupsGandHwhereRDis the set of all functions from a finite setDto a finite setR,Gacts onDandHacts onR. We present an algorithm for obtaining the equivalence classes of functions counted in de Bruijn's theorem, i.e., to determine which functions belong to the same equivalence class. We also use our algorithm to construct the family of non-isomorphicfm-graphs relative to a given group.

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On the cycle index of a product of permutation groups

Journal of Combinatorial Theory ◽

10.1016/s0021-9800(68)80008-0 ◽

1968 ◽

Vol 4 (3) ◽

pp. 277-299 ◽

Cited By ~ 20

Author(s):

Michael A. Harrison ◽

Robert G. High

Keyword(s):

Permutation Groups ◽

Cycle Index

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On Simply Transitive Primitive Permutation Groups

Canadian Journal of Mathematics ◽

10.4153/cjm-1969-117-5 ◽

1969 ◽

Vol 21 ◽

pp. 1062-1068 ◽

Cited By ~ 1

Author(s):

R. D. Bercov

Keyword(s):

Abelian Group ◽

Direct Product ◽

Prime Power ◽

Direct Decomposition ◽

Permutation Groups ◽

Prime Power Order ◽

Schur Ring ◽

Abelian Subgroups ◽

Prime 2

In (1) we considered finite primitive permutation groups G with regular abelian subgroups H satisfying the following hypothesis:(*) H = A × B × C, where A is cyclic of prime power order pα ≠ 4, B has exponent pβ < pα, and C has order prime to p.We remark that an abelian group fails to satisfy (*) (apart from the minor exception associated with the prime 2) if and only if it is the direct product of two subgroups of the same exponent.We showed in (1) that such a group G is doubly transitive unless it is the direct product of two or more subgroups each of the same order greater than 2. This was done by showing that (in the terminology of (3)) the existence of a non-trivial primitive Schur ring over H implies such a direct decomposition of H.

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